On class numbers of quadratic extensions¶over function fields

Kimura, Iwao

doi:10.1007/s002290050087

Cited by 6 publications

(4 citation statements)

References 7 publications

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“…The analogous result for function fields was proven in 1999 by Ichimura [12]. Horie and Onishi [9,10,11], Jochnowitz [14], and Ono and Skinner [29] proved that there are infinitely many imaginary quadratic number fields with class number not divisible by a given prime p. Quantitative results on the density of quadratic fields with class number indivisible by 3 have been obtained by Davenport and Heilbronn [3], Datskovsky and Wright [2], and Kimura [16] (for relative class numbers). Kohnen and Ono made further progress in [17].…”

Section: Introductionmentioning

confidence: 68%

Function fields with class number indivisible by a prime l

et al. 2011

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It is known that infinitely many number fields and function fields of any degree m have class number divisible by a given integer n. However, significantly less is known about the indivisibility of class numbers of such fields. While it's known that there exist infinitely many quadratic number fields with class number indivisible by a given prime, the fields are not constructed explicitly, and nothing appears to be known for higher degree extensions. In [32], Pacelli and Rosen explicitly constructed an infinite class of function fields of any degree m, 3 ∤ m, over F q (T ) with class number indivisible by 3, generalizing a result of Ichimura for quadratic extensions. Here we generalize that result, constructing, for an arbitrary prime ℓ, and positive integer m > 1, infinitely many function fields of degree m over the rational function field, with class number indivisible by ℓ.

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Section: Introductionmentioning

confidence: 68%

Function fields with class number indivisible by a prime l

et al. 2011

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“…The analogous result for function fields was proven in 1999 by Ichimura [14]. Horie and Onishi [11,12,13], Jochnowitz [15], and Ono and Skinner [24] proved that there are infinitely many imaginary quadratic number fields with class number not divisible by a given prime p. Quantitative results on the density of quadratic fields with class number indivisible by 3 have been obtained by Davenport and Heilbronn [6], Datskovsky and Wright [4], and Kimura [17] (for relative class numbers). Kohnen and Ono made further progress in [18].…”

mentioning

confidence: 68%

Indivisibility of class numbers of global function fields

Pacelli

Rosen

2009

Acta Arith.

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“…In [6], Kimura proved that there are infinitely many quadratic extensions F over K such that the class number of F is not divisible by 3. For an odd prime number l, Ichimura [5] constructed infinitely many imaginary quadratic extensions F over K such that the class number of F is not divisible by l, when the order of q mod l in the multiplicative group (Z/lZ) * is odd or l = p.…”

Section: Indivisibility Of Class Numbers Of Imaginary Quadratic Functmentioning

confidence: 99%

Indivisibility of class numbers of imaginary quadratic function fields

Byeon

2008

Acta Arith.

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On class numbers of quadratic extensions¶over function fields

Cited by 6 publications

References 7 publications

Function fields with class number indivisible by a prime l

Function fields with class number indivisible by a prime l

Indivisibility of class numbers of global function fields

Indivisibility of class numbers of imaginary quadratic function fields

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